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Home , Buckminster Fuller, Synergetics (Fuller), Tensegrity

A Fuller Explanation The Synergetic Geometry of R. Collection Series Editor Arthur L. Loeb Department of Visual and Environmental Studies

Amy C. Edmondson A Fuller Explanation: The Synergetic Geometry of R. Buckminster Fuller, 1987

In Preparation Marjorie Senechal Shaping Space: A Polyhedral Approach and George Fleck (Eds.) Arthur L. Loeb Concepts and Images

Amy C. Edmondson

A Fuller Explanation The Synergetic Geometry of R. Buckminster Fuller

A Pro Scientia Viva Title

Birkhauser Boston . Basel . Stuttgart Amy C. Edmondson A Fuller Explanation The Synergetic Geometry of R. Buckminster Fuller

Coden: DSCOED First Printing, 1987

Library of Congress Cataloging in Publication Data Edmondson, Amy C. A Fuller explanation. The synergetic geometry of R. Buckminster Fuller. (Design science collection) "A Pro scientia viva title." Bibliography: p. Includes index. 1. theory. 2. Thought and thinking. 3. Mathematics-Philosophy. 4. Geometry-Philosophy. 5. Fuller, R. Buckminster (Richard Buckminster). 1895- . I. Title. II. Series. Q295.E33 1986 003 86-14791

CIP- Kurztitelaufnahme der Deutschen Bibliothek Edmondson, Amy c.: A Fuller explanation: the synerget. geometry of R. Buckminster Fuller / Amy C. Edmondson-I. print. -Boston; Basel; Stuttgart: Birkhauser, 1986. (Design science collection) (A pro sci entia viva title) ISBN 978-0-8176-3338-7 ISBN 978-1-4684-7485-5 (eBook) 001 10.1007/978-1-4684-7485-5

Frontispiece photograph courtesy of Phil Haggerty.

© Birkhauser Boston, Inc., 1987 Softcover reprint of the hardcover 1st edition 1987 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recorded or otherwise, without prior written permission of the copyright owner, Birkhauser Boston, Inc., 380 Green Street, P.O.B. 2007, Cambridge, MA 02139, U.S.A.

ISBN 978-0-8176-3338-7

Typeset by Science Typographers, Inc., Medford, New York. To my parents Mary Dillon Edmondson and Robert Joseph Edmondson Contents

Series Editor's Foreword...... xv Preface...... XIX Acknowledgments ...... xxi Note to Readers ...... XXlll Introduction...... xxv

1. Return to Modelability ...... 1 From Geometry to Geodesics: A Personal Perspective...... 3 Operational Mathematics ...... 6 Experimental Evidence ...... 7 Nature's Coordinate System...... 9 Universe...... 11 Generalized Principles...... 12 Return to Mode1ability ...... 14

2. The Irrationality of Pi ...... 15 "Nature Isn't Using Pi" ...... 16 Finite Accounting System ...... 18 Which Way Is "Up"? ...... 19 Visual Literacy ...... 22 Peaceful Coexistence ...... 23

3. and Synergy ...... 25 Conceptual and Real Systems ...... 28 Limits of Resolution as Part of the Whole-Systems Approach ...... 30 Synergy...... 33 x Contents

4. Tools of the Trade ...... 36 Plato's Discovery ...... 37 Triangles ...... 38 Squares ...... 40 Pentagons...... 41 A Limited Family ...... 41 Euler's Law ...... 43 Duality ...... 45 Truncation and Stellation ...... 46 An Experiment...... 48 "Intertransformability" ...... 51 Symmetry ...... 52

5. Structure and "Pattern Integrity" ...... 54 Pattern Integrity...... 56 Structure ...... 60

6. Angular Topology ...... 65 Frequency and Size...... 66 Units of Measurement ...... 66 Time and Repetition: Frequency versus Continuum ...... 67 Topology and Vectors...... 68 Vector Polyhedra...... 68 Dimension ...... 70 Size...... 72 Planes of Symmetry...... 73 Other Applications of Dimension ...... 74 Angular Topology...... 75 Angular Takeout: An Example ...... 78 Angle Types ...... 79

7. Vector Equilibrium...... 82 "Nature's Own Geometry" ...... 84 Spatial Considerations ...... 85 Planar Equilibrium ...... 87 as Vector Equilibrium ...... 90 VE: Results ...... 91 Degrees of Freedom ...... 93

8. Tales Told by the Spheres: Closest Packing ...... 100 Equilibrium: Equalization of Distances ...... 100 Symmetry versus Specificity of Form...... 101 Contents xi

Organization of Identical Units...... 101 New Level of Focus...... 102 Background: Closepacking ...... 102 Planes of Symmetry...... 106 Fuller Observations ...... 109 Tetrahedra ...... 110 Vector Equilibrium ...... 114 Frequency ...... 114 Icosahedron ...... 117 Further Discoveries: Nests...... 120 "Interprecessing" ...... 121 A Final Philosophical Note ...... 124

9. Isotropic Vector Matrix ...... 127 A Quick Comparison: " Accounting" ...... 130 Cells: "Inherent Complementarity" ...... , 131 A Complete Picture ...... 133 Angles...... 134 Locating New Polyhedral Systems...... 135 Duality and the IVM ...... 136 Angles...... 136 Polyhedra ...... 137 Domain...... 138 Framework of Possibility...... 140 Invention: Octet ...... 141

10. Multiplication by Division: In Search of Cosmic Hierarchy. .. 143 Volume...... 143 Results: Volume Ratios...... 144 Shape Comparisons: Qualities of Space...... 146 Volume: Direct Comparison...... 147 Multiplication by Division...... 149 as Starting Point...... 149 Cube...... 150 Vector Equilibrium ...... 152 Rhombic Dodecahedron ...... 153 Multiplication by Division ...... 154 Cosmic Hierarchy (of Nuclear Event Patternings) ...... 157 Volume Reconsidered ...... 157

11. Jitterbug...... 159 Folding a Polyhedron ...... 160 Volume and Phase Changes ...... 163 xii Contents

Icosahedron...... 163 Single Layer versus IVM ...... 164 "Trans-Universe" versus "Locally Operative" ...... 165 Fives...... 165 "S-Modules" ...... 167 Icosahedron and Rhombic Dodecahedron ...... 167 Pentagonal Dodecahedron...... 168 Four Dimensions ...... 169 Complex of Jitterbugs ...... 170 Other Dynamic Models ...... 172 Topology and Phase ...... 172

12. Ii All-Space" Filling: New Types of Packing Crates ...... 175 Plane Tessellations ...... 176 Filling Space ...... 177 Complementarity ...... 178 Other Space Fillers ...... 179 The Search Continues ...... 180 The Dual Perspective...... 181 Duality and Domain in Sphere Packing...... 183 Truncated ...... 184 Two to One: A Review ...... 185

13. The Heart of the Matter: A- and B-Quanta Modules...... 189 A-Quanta Modules ...... 190 B-Quanta Modules ...... 190 Energy Characteristics...... 193 Mite ...... 195 Mirrors ...... 197 Cubes into Mites...... 198 Rhombic Dodecahedra ...... 198 Coupler ...... 199 Volume and Energy ...... 201 Review: All-Space Fillers...... 203

14. Cosmic Railroad Tracks: Great Circles ...... 206 Why Are We Talking About Spheres? ...... 207 New Classification System ...... 208 Great-Circle Patterns ...... 209 Least Common Denominator ...... 213 LCD: "Intertransformability" ...... 215 LCD of 31 Great Circles ...... 216 VE's 25 Great Circles...... 217 Contents xiii

Operational Mathematics ...... 219 Conservation of Angle...... 221 Foldable Systems ...... 223 Energy Paths ...... 226 Gas Molecules ...... 226 Great-Circle Railroad Tracks of Energy...... 228 Icosahedron as Local Shunting Circuit...... 229 Inventory: Seven Unique Cosmic Axes of Symmetry ...... 230 Excess of One ...... 230

15. From Geodesic to : The Invisible Made Visible. . .. 232 Theory Behind Geodesic Structures: Summary...... 235 Geodesic Design in Nature...... 237 Geodesic : Design Variables...... 240 Tensegrity ...... 243 Nature's Example ...... 245 New Concept of Construction...... 249 Modeling the Invisible ...... 250 Tensegrity Polyhedra...... 251 Pneumatics...... 255 Case in Point: Donald Ingber...... 257

16. "Design Science" ...... 258 "Comprehensive Anticipatory Design Science" ...... 258 "Comprehensive ... " ...... 259 " ... Anticipatory ... " ...... 261 Map...... 263 Suspended Storage Systems ...... 266 More with Less...... 268

Appendices...... 271 A. Trigonometric Calculations ...... 271 B. Volume Calculations for Three Prime Structural Systems ...... 273 C. Sources of Additional Information...... 277 D. Special Properties of the Tetrahedron...... 279 E. Glossary ...... 281

Bibliography ...... 287 Notes...... 289 Index...... 295 Series Editor's Foreword

In a broad sense Design Science is the grammar of a language of images rather than of words. Modern communication techniques enable us to transmit and reconstitute images without the need of knowing a specific verbal sequential language such as the Morse code or Hungarian. International traffic signs use international image symbols which are not specific to any particular verbal language. An image language differs from a verbal one in that the latter uses a linear string of symbols, whereas the former is multidimensional. Architectural renderings commonly show projections onto three mutually perpendicular planes, or consist of cross sections at differ• ent altitudes representing a stack of floor plans. Such renderings make it difficult to imagine buildings containing ramps and other features which disguise the separation between floors; consequently, they limit the creativity of the architect. Analogously, we tend to analyze natural structures as if nature had used similar stacked renderings, rather than, for instance, a system of packed spheres, with the result that we fail to perceive the system of organization determining the form of such structures. Perception is a complex process. Our senses record; they are analogous to audio or video devices. We cannot claim, however, that such devices perceive. Perception involves more than meets the eye: it involves processing and organization of recorded data. When we classify an object, we actually name an abstract concept: such words as octahedron, collage, tessellation, ; each designates a wide variety of objects sharing certain characteristics. When we devise ways of transforming an octahedron, or determine whether a given shape will tessellate the plane, we make use of these characteristics, which constitute the grammar of structure. The Design Science Collection concerns itself with various aspects of this grammar. The basic parameters of structure, such as symme- xvi Series Editor's Foreword try, connectivity, stability, shape, color, size, recur throughout these volumes. Their interactions are complex; together they generate such concepts as Fuller's and Snelson's tensegrity, Lois Swirnoff's mod• ulation of surface through color, self-reference in the work of M. C. Escher, or the synergetic stability of ganged unstable polyhedra. All of these occupy some of the professionals concerned with the com• plexity of the space in which we live, and which we shape. The Design Science Collection is intended to inform a reasonably well• educated but not highly specialized audience of these professional activities, and particularly to illustrate and to stimulate the interac• tion between the various disciplines involved in the exploration of our own three-dimensional, and in some instances more-dimen• sional, spaces. When R. Buckminster Fuller recalled his days as a schoolboy in Milton, Massachusetts, he related how his mathematics teacher would introduce two-dimensional surfaces by placing lines of zero thickness side by side; young Buckminster used to wonder how one could create a finite surface out of nothing. Similarly, he could not accept the stacking of planes of zero thickness to create volumes. Intuitively, he sensed that areas and volumes are as different from each other as are forces and velocities: one cannot mix quantities of different dimensionality. Accordingly, Fuller learned to compare three-dimensional objects with each other, and hence to add, sub• tract, and transform them from and into each other rather than creating them out of objects of lower dimensionality. In doing so he came to discard the conventional orthogonal system which has blinded architects as well as solid-state scientists, and followed natural structure in designing his stable light-weight structures. Two days before Harvard Commencement in 1983, Amy Edmond• son called me from Buckminster Fuller's office in , saying that Fuller had decided at the last moment to attend the Commencement exercises, and wondered whether I might still be free to have dinner with them the following evening. Amy had graduated from Harvard with combined honors in Applied Science and in Visual and Environmental Studies, and had been working for Fuller since then. At dinner we planned a working session in August at Fuller's island off the Maine coast. Unfortunately that Commence• ment turned out to be Buckminster's last, and when I saw Amy again it was at the combined service in memory of Buckminster and Anne Fuller. We decided right then and there that the best tribute would be a volume aiming at translating Buckminster Fuller's ideas Series Editor's Foreword xvii and idiom into a language more accessible to the lay audience and more acceptable to the scientist. Amy Edmondson has succeeded admirably in conveying to us not just the idiom but also the atmosphere of Fuller's "office." There were no professional draftsmen, for the staff was minimal. We believe that the sense of a direct link to the Fuller office would be enhanced by reproducing Edmondson's own illustrations directly, just as she would have produced them there. With A Fuller Explanation we initiate the Design Science Collec• tion, an exploration of three-dimensional space from the varied perspectives of the designer, artist, and scientist. Through this series we hope to extend the repertoire of the former to professions by using natural structure as an example, and to demonstrate the role of esthetic sensibility and an intuitive approach in the solution of scientific problems.

ARTHUR L. LOEB Cambridge, Massachusetts Preface

Buckminster Fuller has been alternately hailed as the most innova• tive thinker of our time and dismissed as an incomprehensible maverick, but there is a consistent thread running through all the wildly disparate reactions. One point about which there is little disagreement is the difficulty of understanding Bucky. "It was great! What did he say?" is the oft-repeated joke, describing the reaction of a typical enraptured listener after one of Fuller's lectures. Not surprisingly then, Fuller's mathematical writing has not at• tracted a mass audience. Rather, synergetics has become a sort of Fuller proving ground, into which only a few scientific-minded types dare to venture. "Oh, I'll never be able to understand it then" has been the response of countless people upon learning that the subject of my book is synergetics. This reaction would have saddened Bucky immensely: he was so sure his geometry was appropriate for five• year-olds! However, such shyness is understandable; deciphering Fuller's two volumes, Synergetics and Synergetics 2, requires a sizable commitment of time and patience from even the most dedi• cated reader. Study groups have gone a long way toward helping individuals unravel the idiosyncratic, hyphenated prose of these two works, but the task, still arduous, is not for everyone. However, as those who dared it will have discovered, the major concepts pre• sented in Fuller's intimidating books are not inherently difficult, and much of synergetics can be explained in simple, familiar terms. That is the purpose of A Fuller Explanation. Synergetics, in the broadest terms, is the study of spatial complex• ity, and as such is an inherently comprehensive discipline. Designers, architects, and scientists can easily find applications of this study in their work; however, the larger significance of Fuller's geometry may be less visible. Experience with synergetics encourages a new way of approaching and solving problems. Its emphasis on visual and xx Preface spatial phenomena combined with Fuller's holistic approach fosters the kind of lateral thinking which so often leads to creative breakthroughs. A Fuller Explanation is geared to readers with no mathematical background, but of course it can be read at many levels. Even if one is familiar with some of the concepts, Fuller's unique interpretation and development of them will be enlightening. This book should appeal to anyone interested in patterns and design and how things work. Synergetics is also fascinating as a reflection of Fuller himself; his wide-eyed appreciation of nature and human invention alike exudes from his expression of these geometric concepts. The primary pur• pose of this volume is thus to present the nuts and bolts of synerget• ics, the tools with which to continue exploring this discipline; but almost as importantly, 1 hope to convey the spirit of Fuller's inquiry into the organizing principles of nature. When Bucky Fuller looked around, he saw, not trees and roads and butterflies, but a miraculous web of interacting patterns. As he describes these patterns, using his peculiar blend of antiquated phrases and electronic-age jargon, one cannot help concluding that no child was ever as startled as the young Bucky to discover that the world is not what it appears, that the apparently solid and lifeless rock is a bundle of energetic atomic activity. He never lost that awe. Synergetics is his attempt to give some of it away. 1 have included (along with explanations and definitions) many of Fuller's own descriptions and invented terms. Although 1 set out to interpret and explain Bucky in ordinary language-a task 1 found increasingly tricky as 1 became more deeply involved in., the project -I began to sense a deep appropriateness to Bucky's peculiar phraseology. Many passages that seem convoluted at first reading later seem to express his meaning more precisely than could any substitutes. As 1 quote Fuller often in this book, readers will be able to judge for themselves. Acknowledgments

Arthur Loeb first introduced me to the intricate structure of ordered space through his wonderful course at Harvard University. As editor of this book, he generously resumes his instructive role-seven years later, long after his responsibility to a former student has expired. For this I am deeply grateful. Dr. Loeb has perfected the art of gentle criticism, and it was an extraordinary privilege to have had this exposure to his kindness and wisdom. I am greatly indebted to Bucky's family, especially and Jaime Snyder, for their enthusiastic endorsement of my decision to write this book; their commitment to making available information about Bucky's work and life has enabled A Fuller Explanation to exist. I also want to thank John Ferry at the Buck• minster Fuller Institute for his assistance in locating photographs. I am grateful to many people for their encouragement and inspira• tion, but above all to my parents, Bob and Mary Edmondson, whose support of this project was indispensable. I want to thank Martha Lerski for her eager and skillful editing; her help was invaluable. Also, I profited enormously from the insights of many who have attended synergetics workshops, but to none am I more indebted than to Darrell Mullis, Tony Perry, Martha Stires Perry, and Carrie Fisher. My husband, Mark Carpenter, provided the constant sustenance that nurtured both this book and its author throughout the writing process. He contributed many of the drawings, hours of editing, and most importantly his inexhaustible intellectual curiosity. Finally, Bucky was and remains an endless source of inspiration. He was the best teacher and friend any 21-year-old just entering the "real world" could have had. It would be redundant to try to express my gratitude for this experience here, for that is what I hope this book will do.

AMY C. EDMONDSON New York. New York Note to Readers

Quotations from Fuller's Synergetics: Explorations in the Geometry of Thinking, (which is organized into numbered sections and subsec• tions in such a way that each paragraph is given its own reference number) are followed by the numerical section reference in parenthe• ses. Passages from Synergetics 2: Further Explorations in the Geometry of Thinking will be referenced likewise, with the addition of the letter "b" after the section number. Quotations are reprinted with permission of Macmillan Publishing Company, New York, from Synergetics: Explorations in the Geome• try of Thinking by R. Buckminster Fuller, in with E.J. Applewhite. Copyright ©1975 by R. Buckrninster Fuller and from Synergetics 2: Further Explorations in the Geometry of Thinking by R. Buckminster Fuller, in collaboration with E.J. Applewhite. Copyright ©1979 by R. Buckminster Fuller. Photograph courtesy of ARGO Construction, Inc., , Quebec, Canada. Introduction

The scene is Montreal, 1967: travelers from around the world emerge from a subway station at the Expo site, and catch their first glimpse of an enormous transparent bubble. Looking and exclaiming, they gravitate toward this strange monument, which is the United States Pavilion, and few notice the stocky white-haired old man, straining his slightly deaf ears to glean their reactions. Buckminster Fuller, playing the disinterested bystander, along with Anne Hewlett Fuller, his wife of exactly fifty years, is a triumphant eavesdropper; the candid observers have enthusiastically approved his design. Fifteen years later, he recalls that summer morning with a playful grin, clearly enjoying the memory of his short stint as detective, and I can almost see him there, standing next to Anne, silently sharing the knowledge that the years of perserverence-ignoring skepticism and often decidedly harsh disapproval of his mathematical work-were vindicated. Bucky is such a gifted story teller that I also imagine I can see the huge geodesic sphere reflecting the intense summer sun, and it looks more like one of nature's creations than architecture. But it is steel, Plexiglas and human ingenuity that have created this glittering membrane, which was, in 1967, the world's largest dome, spanning 250 feet and reaching an altitude equal to that of a twenty-story building, without any interior support. More than just the millions who visited Expo '67 have admired this architectural feat, and humanity has found countless other uses for the , as evidenced by the 100,000 such structures of various materials and sizes that are sprinkled around the globe. However, the "synergetic geometry," which lies behind Fuller's remarkable design, has remained almost completely obscure. The goal of this book is to catalyze a process which I hope will continue and expand on its own: to rescue Fuller's fascinating material from its unfortunate obscurity.

A. C. E. Experience has shown repeatedly that a mathematical theory with a rich internal structure generally turns out to have significant implica• tions for the understanding of the real world, often in ways no one could have envisioned before the theory was developed.

William P. Thurston and Jeffrey R. Weeks The Mathematics of Three-Dimensional Manifolds Scientific American, July 1984.